Chapter 4.7, Problem 14PPS

To determine

To find: The inverse of the function $f\left(x\right)=25+4x$ .

Answer to Problem 14PPS

The inverse of the function $f\left(x\right)=25+4x$ is $f^{-1}\left(x\right)=\frac{1}{4}x-\frac{25}{4}$ .

Explanation of Solution

Given:

The function is $f\left(x\right)=25+4x$ .

Concept used:

To obtain the inverse of any function $f\left(x\right)=ax+b$ , following steps are used:

Replace $f\left(x\right)$ with $y$ .

Interchange $x$ and $y$ .

Solve for $y$ .

Again, replace $y$ with $f\left(x\right)$ .

Calculation:

Replace $f\left(x\right)$ with $y$ for the function $f\left(x\right)=25+4x$ as follows:

  $y=25+4x$

Interchange $x$ and $y$ in the above expression.

  $x=25+4y$

Solve for $y$ in expression $x=25+4y$ as follows:

  $\begin{array}{c}x-25=4y\\ y=\frac{x-25}{4}\\ y=\frac{1}{4}x-\frac{25}{4}\end{array}$

Replace $y$ with $f^{-1}\left(x\right)$ in above equation to obtain the inverse of the function.

  $f^{-1}\left(x\right)=\frac{1}{4}x-\frac{25}{4}$

Thus, the inverse of the function $f\left(x\right)=25+4x$ is $f^{-1}\left(x\right)=\frac{1}{4}x-\frac{25}{4}$ .